Table of Links

1. Abstract & Introduction
2. Notation and outline of the proof
3. The set Ψ1
4. Zeros of L(s, ψ)L(s, χψ) in Ω
5. Some analytic lemmas
6. Approximate formula for L(s, ψ)
7. Mean value formula I
8. Evaluation of Ξ11
9. Evaluation of Ξ12
10. Proof of Proposition 2.4
11. Proof of Proposition 2.6
12. Evaluation of Ξ15
13. Approximation to Ξ14
14. Mean value formula II
15. Evaluation of Φ1
16. Evaluation of Φ2
17. Evaluation of Φ3
18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums

References

References

[1] R. Balasubramanian, J. B. Conrey and D. R. Heath-Brown, Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial, J. reine angew. Math. 357(1985), 161-181.

[2] J. B. Conrey, A. Ghosh and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16(1984), 421-424.

[3] J. B. Conrey, A. Ghosh and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, Number theory, trace formulas and discrete groups (Academic Press, Boston, 1989), 185-199.

[4] J. B. Conrey, A. Ghosh and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3)76(1998), 497-522.

5] J. B. Conrey, H. Iwaniec and K. Soundararajan, Asymptotic large sieve, Preprint 2011, arxiv 1105.1176

[6] J. B. Conrey, H. Iwaniec and K. Soundararajan, Critical zeros of Dirichlet Lfunctions, Preprint 2011, arxiv 1105.1177

[7 ] H. Davenport, Multiplicative Number Theory, 3rd. ed. (revised by H. L. Montgomery), Springer-Verlag, New York, 2000

[8] D. Goldfeld, A simple proof of Siegel’s theorem, Proc. Nat. Acad. Sci. USA 71(1974), 1055.

[9] D. Goldfeld, An asymptotic formula relating the Siegel zero and the class number of quadratic fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(1975), 611-615.

[10] D. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(1976), 624-663.

[11] A. Granville and H. M. Stark, ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant, Invent. Math. 139(2000), 509-523.

[12] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math., 84(1986), 225-320.

[13] D. R. Heath-Brown, Small class numbers and the pair correlation of zeros, Conference in Celebration of the Centenary of the the Proof of the Prime Number Theorem (A Symposium on the Riemann Hypothesis), Seattle, 1996.

[14] H. Iwaniec and E. Kowalski, Analytic number theory, Coll. Pulb. vol. 53, Amer. Math. Soc. 2004

[15] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic Lfunctions and Landau-Siegel zeros, Israel J. Math., 120(2000), 155-177.

[16] A. Karatsuba, Basic Analytic Number Theory, (translated from Russian by M. Nathanson), Springer-Verlag, New York, 1993.

[17] H. L. Montgomery, The pair correlation of zeros of the zeta-function, Analytic number theory, Proc. Symp. Pure Math. Vol 24, 181-193, (Amer. Math. Soc., Providence, RI. 1973)

[18] C. L. Siegel, Uber die Classenzahl quadratischer Zuhlk¨orper, ¨ Acta Arith. 1(1936), 83-86.

[19] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd. ed. (revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986